Question

Consider an online store where a number of customers visit and
buy a product every hour. Let *X* be the number of people
who enter the store per hour.

The store is active for 14 hours per day, every day of the week. It is calculated from data collected that the average number of customers per hour is 10.

**(a)** When is it appropriate to
approximate a Poisson Distributed random variable with a Normal
Distribution? State the appropriate parameters for the Normal
Distribution.

.

**(b)** Calculate the probability that 6
or less customers enter the store. *Hint: Remember to use the
correction for continuity and R to help calculate
the probability.*

*.*

**(c)** The manager of the store decides
to open up a chain of five stores and it is observed that the
average number of customers entering the stores is the same. What
is the mean and variance of the distribution for the average of all
the five stores?

**.**

**(d)** Calculate the probability for the
average of the five stores when 6 or less customers enter?

Answer #1

a) If λ is greater than about 10, then the normal distribution
is a good approximation if an appropriate continuity correction is
performed. The approximate Normal distribution has a **mean
of 10 and variance 10**.

b) The probability that 6 or less customers enter the store =
**0.1342.**

The R code is *>pnorm(6.5, 10,
sqrt(10)*

c) As the sum of poison distribution with the parameter nλ thus
the mean=10*5 = 50 and the variance is 10*5=50 for the average sum
of all the five stores. Thus for the average distribution
**mean is 10 and the variance is 2.**

d) The probability for the average of the five stores when 6 or
less customers enter is **0.00666**

The R code is *>pnorm(6.5, 10,
sqrt(2)*

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